Problem: If $\cos \theta = \frac{2}{3},$ then find $\cos 2 \theta.$
Explanation: From the double angle formula,
\[\cos 2 \theta = 2 \cos^2 \theta - 1 = 2 \left( \frac{2}{3} \right)^2 - 1 = \boxed{-\frac{1}{9}}.\]